Convergence Speed in Formation Control of Multi-Agent Systems - A Robust Control Approach

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1 nd IEEE Conference on Decision and Control December -,. Florence, Italy Convergence Speed in Formation Control of Multi-Agent Systems - A Robust Control Approach Ulf Pilz and Herbert Werner Abstract In this work a robust design approach to improve the convergence speed in formation control of multi-agent systems by simultaneously shaping the control feedback law and the weights of the interconnection structure is proposed. The communication topology is assumed to be fixed and undirected. It is shown that for agents modeled as discrete-time doubleintegrators, convergence speed of the proposed method almost reaches the theoretical optimum. The convergence speed can be further increased if the feedback control law is allowed to be dynamic. Although the number of decision variables in this approach grows linearly with the size of the weighting matrix, the synthesis method is applied also to a more realistic linearized model of a quad-rotor helicopter, where the simulation results indicate that this approach is reasonable for a sufficiently large number of high-order agents. I. INTRODUCTION This paper is concerned with the synthesis of formation control laws for distributed control of multi-agent systems (MAS). A MAS consists of two or more identical agents which are physically decoupled but share a common objective, and which are able to communicate with each other (see, e.g., [], []). The communication topology is assumed to be known; it is modeled as an undirected graph and can be described by the graph Laplacian matrix L. Here we are interested in decentralized formation control schemes for multi-agent systems as proposed in [], []. Such formation control schemes can be interpreted in terms of consensus and control. Methods for improving the speed of convergence in such schemes by individually weighting the communication links have been proposed in the literature (see references discussed below). In this paper we propose a way to find simultaneously the optimal weighting matrix and the optimal feedback law for fast convergence speed while taking into consideration additional constraints like disturbance rejection or restrictions on the control effort. Without relying on results that depend on a decomposition of the system (see [], [] [7]), a method is proposed that minimizes the H norm of a certain transfer function while at the same time determining the weights of the communication topology. Convergence speed of discrete-time single-integrator agents has been considered in [8], [9], where [8] considered an undirected and fixed communication topology to determine the fastest convergence speed for single integrator U. Pilz and H. Werner are with the Institute of Control Systems, Hamburg University of Technology, Eissendorfer Str., D-7 Hamburg, Germany. {ulf.pilz, h.werner}@tu-harburg.de. Corresponding author, Tel.: agents. Directed and time-varying communication topologies were considered in [9] and a polynomial-time averaging algorithm was derived. Stability analysis of continuoustime and discrete-time double integrator consensus for fixed and switching communication topologies was performed in [], []. Relevant results for convergence speed of double integrator agents are reported in [], where the parameters of the feedback law are optimized while having predefined weights on the communication links. Unlike [], where an analytic global optimal solution for the fastest convergence speed of a discrete-time double integrator model is obtained, the approach taken in this paper is more flexible in that it can be used to design dynamic formation control schemes for multi-agent systems with general LTI agent models. The main contribution of this work is the presentation of a synthesis method for a simultaneous design of a formation control law and a weighting matrix W for the communication links to achieve fast convergence. Furthermore, in the mixedsensitivity framework employed here other relevant control objectives are taken into account like disturbance rejection or limits on the control input. Since restrictions on the structure and the order of the controller lead to a nonsmooth, nonconvex optimization problem, algorithms of the package HIFOO for fixed structure controller synthesis are utilized, see []. The numerical complexity of this approach is not independent of the number of agents; the number of decision variables grow linearly with the size of the weighting matrix, i.e., the number of links in the communication topology. This paper is organized as follows. Section II briefly recalls the graph-theoretical framework and introduces the agent models. In Section III, the synthesis method that allows for a simultaneous design of a formation control law and a weighting matrix is proposed. Numerical results for a doubleintegrator model and more general higher-order LTI agents are presented in Section IV. Finally, conclusions are drawn and an outlook to future work is given in Section V. Notation: R + is the set of all positive real numbers, Z is the set of integers and N is the set of non-negative integers; I q denotes the q q identity matrix; A T denotes the transpose of a matrixa. The sets ofp q real and complex matrices are R p q and C p q, respectively. is the Kronecker product. The discrete-time index is denoted as t Z and all signals are assumed to be zero for t <. Furthermore, Q (m) is a shorthand notation for Q I m and Q = I N Q, where Q may represent either a matrix or a dynamic system. The open and closed unit disks are defined as D = {ζ ζ C, ζ < } and D = {ζ ζ C, ζ }, respectively. RH is the space of proper and real-rational transfer functions having all //$. IEEE 667

2 poles inside the open unit disk D. F L (P(z),K(z)) denotes the lower linear fractional transformation (LFT) ofp(z) with K(z). II. PRELIMINARIES In this section we will briefly describe the graph theoretical background needed to describe a model of a MAS and describe the double-integrator consensus protocol. A. Graph Theory Concepts from graph theory are commonly used to model communication and sensing topologies in a MAS. First, we recall a few basic definitions (see, e.g., []). An undirected graph G is completely described by a vertex set V, an edge set E and a weighting matrix W, where V = {v,...,v N }, E V V and W = diag(w,...,w m ) R m m. The number of vertices is N N and the number of edges in the graph is m N. The edge set is defined as E {e ij = (v j,v i ) : v i,v j V}. For undirected graphs the equation e ij = e ji holds. We assume that there are no selfloops, i.e.,(i,i) / E,i =,...,N. If(v j,v i ) is an edge ofg, then we say that v j is a neighbor of v i. The set of neighbors of vertexv i is defined asn i (G,v i ) = {v j : (v j,v i ) E(G)}. The adjacency matrix A = [a ij ] R N N associated with G is defined such that a ij = a ji has a positive value if (v j,v i ) E(G) and a ij = a ji = otherwise. The weighted graph Laplacian is given as L = DWD T, where D R N m indicates the oriented incidence matrix, where the rows and columns of the matrix are indexed by the vertices and edges, respectively. The element [d ik ] of D has the value + if node v i is the tail of edge e k, if node v i is the head node of edge e k and otherwise. It is important to note that the resulting undirected weighted graph Laplacian matrix L does not depend on the choice of the orientation of the edges for the incidence matrix D. Due to its construction the Laplacian has at least one zero eigenvalue with an associated eigenvector and from Geršgorin s circle theorem it follows that all nonzero eigenvalues are in the open right half plane. B. Higher-Order Agents A group of N identical agents is considered, each agent represented as a feedback loop comprising a linear discretetime model P(z) with a state-space representation x i (t+) = Ax i (t)+bu i (t) y i (t) = C f x i (t) φ i (t) = C l x i (t), where i =,,..., N, and a local controller K(z) shown in Fig. with state-space representation ψ i (t+) = A K ψ i (t)+b K e i (t)+b K φ i (t) u i (t) = C K ψ i (t)+d K e i (t)+d K φ i (t). Here x i R n are the states of the plant; ψ i R nk are the controller states,u i R h are the control inputs;φ i R m are measured outputs that are available to agent i only; y i R p are measured outputs that are communicated to other agents () () e i H(z) K(z) Fig.. u i Single agent P(z) and e i R p is the control error of agent i. The controller K(z) of agent i receives the control error e i and local feedback φ i. It is assumed that P(z) is stabilized by the local feedback signal φ i which is not transmitted to other agents in the formation. Therefore, the transfer function H(z) from e i to y i, denoted by H(z) = F L (P(z)K(z), I m ), is assumed to have all eigenvalues inside the unit disc D. In the closed-loop MAS, the communication structure is modeled by the weighted graph Laplacian matrix L (p) = D (p) W (p) D(p) T which is adapted to the number of outputs p. The generic closed-loop representation of a MAS consisting of higher order agents and weighted undirected communication links is shown in Fig., see [6]. r η L (p) Fig.. e Ĥ(z) Block diagram of a MAS [ ] The reference input r = r T r T... rn T T determines the shape of the formation. The absolute error between the output y = [ y T y T... yn] T T and the reference input r is given by η = [ T, η T η T... ηn] T whereas the relative error is defined as e = L (p) η, where e = [ e T e T... e T N] T. For signal dimensions we have r,η,e,y R Np and u = [ u T u T... u T N] T R Nh. C. Second-Order Consensus Protocol As a special case which has received considerable attention in the literature, a second order consensus protocol is considered in detail. Each node of the graph is modeled as a second order discrete-time double integrator of the form ξ i (t+) = ξ i (t)+τζ i (t) ζ i (t+) = ζ i (t)+τu i (t), i =,...,N, where τ N is the sampling interval, and ξ i R and ζ i R are the states of the double integrators. The following consensus protocol is considered in this paper, see [] u i (t) = φ i y y i () N a ij [(ξ i (t) ξ j (t))+α(ζ i (t) ζ j (t))], () j= where u i (t) R is the consensus feedback control law and α R is a tuning parameter. Note that ()-() is a distributed protocol since local updates only depend on agents from 668

3 which agent i receives information and are thus neighbors of agent i. The update for the whole MAS is given as [ ] [ ][ ] [ ] ξ(t+) IN τi = N ξ(t) + u(t) () ζ(t+) I N ζ(t) τ }{{}}{{}}{{}}{{} x(t+) x(t) or  [ ξ(t) u(t) = [ I N αi N ] L }{{} () ζ(t) K [ ] [ ][ ] ξ(t+) IN τi = N ξ(t) ζ(t+) τl I N ταl ζ(t), }{{} where ξ(t) = [ ξ (t) T ξ (t) T... ξ N (t) T] T, and ζ(t), u(t) are defined accordingly. Furthermore, the matrices A, B, K and Ξ are adopted to the number of agents N, i.e.,  = I N A and the Laplacian matrix L is adopted to the dimension of the output signal p =, i.e., L () = L I. Note that the communication topology is assumed to be fixed and the sampling time τ is known in advance. Furthermore, the only parameter which directly influences the static feedback law K is α. The transformation of the double integrator model into frequency domain gives the following transfer function representation [ ] ( τ P(z) = zi N Â) B = IN (8) Ξ B ], (z ) τ z Note that a MAS with double integrator agents is a special case of the system shown in Fig., where dim (y i ) =, dim (φ i ) = and C f = I. The control input is determined as u = Ke with K restricted to be static. Furthermore, η = L () e = L () y. A convergence analysis for double integrator dynamics has been performed in [] and is summarized in the following lemma: Lemma (cf. []): If the graph G has a spanning tree and all eigenvalues of Ξ (cf. 7) that are not one, have a modulus smaller than one, then the consensus protocol () will achieve consensus asymptotically. Proof: See []. III. CONTROLLER SYNTHESIS This section explains how to construct a controller considering H performance criteria with a gradient-based controller design method. A. Synthesis Problem This section presents the simultaneous design of an H optimal feedback control law () and of the weights w i, i =,...,m of the communication structure. Two applications are considered in this work: the minimization of the convergence time of a MAS consisting of double-integrator models, and the simultaneous synthesis of the weighting matrix and dynamic controllers of any desired order for MAS consisting of high-order agents (). The performance requirements are defined using a mixed-sensitivity approach. Fig. shows the. (6) (7) generalized plant augmented with a sensitivity shaping filter Ŵ S (z) and a control sensitivity shaping filter Ŵ K (z). The w P u Ĝ(z) P(z) Fig.. y η K(z) W (p) Generalized plant Ŵ K (z) Ŵ S (z) D T (p) D (p) exogenous inputs are w P = r and the performance outputs are given as z P = [ z T K zt S] T Referring to Fig., the control problem is formulated as follows: e z K z S min stabilizing K(z),W (p) T zp w P (z), (9) where T zp w P (z) denotes the transfer function from w P to z P. Note that the sensitivity filter ŴS(z) imposes a penalty on the absolute error signal η. Penalties on the relative formation error e will cause the minimization algorithm to set the weighting matrix to a zero. B. Synthesis Method Standard H tools can not be used to solve (9) since the structure of K(z) = IN K(z) cannot be imposed in the synthesis procedure. One way to solve this minimization problem is to use optimization methods able to solve nonconvex problems like neural networks [7], genetic algorithms [8] or gradient-based methods [], [9]. Common to all these algorithms is the generality of the approach since they handle also heterogeneous MAS at the expense of computing the H norm of the closed-loop system and therefore are useless for MAS with a large number of agents. The drawbacks of the direct methods are an excessive computational load and no guarantee of reaching even a local optimum. In this work, a gradient-based optimization method is applied, which is able to handle low-order and fixed-structure controller synthesis problems [] and which has been extended to handle discrete-time and replicated structures as well, for details see [], []. In order to apply the gradientbased method for replicated controller structures, the derivative of the objective function with respect to the decision variables, which correspond to the entries of the state-space matrices of the controller in (), must be calculated. For this purpose, first the gradient of the objective function (9) with 669

4 respect to K(z) = I N K(z) and then with respect to K(z) is needed. The former has already been discussed in [], [9]. The latter has been reported in [] and is exemplarily given here for the controller output matrix C K Lemma (cf. []): The gradient of the closed-loop H norm f = T zp w P (z) with respect to element lk of the output matrix C K R h nk is CKlk f = hn i= n KN j= P ij f Ĉ K ij, () where P = I N ( e l e T k), el denote column l of I h and e k column k of I nk. Note that there is no restriction in considering only static output feedback controllers, since every fixed order dynamic controller design can be seen as a special case of a static controller design problem, see []. IV. SIMULATION RESULTS In the first part of this section, it is shown how the convergence speed of the discrete-time double integrator MAS is influenced by the design method proposed in Section III. For this purpose, a comparison with the optimal solution to this problem which has been analytically determined in [] will be performed. The second part shows how the proposed synthesis method can also be applied to MAS consisting of more general MIMO agents. A. Convergence Speed of Double Integrator Consensus The convergence time of the double-integrator consensus problem is considered. For this purpose, a network with six nodes and fixed communication topology consisting of eight edges is considered, where D = and the sampling interval τ =. Fig. shows the positions η(t), the velocities ζ(t) and the control inputs u(t) for the double integrator model. The advantage of the robust control approach proposed here is the flexibility of minimizing different control objectives like convergence time or control input at the same time and the possibility to synthesize dynamic feedback control laws. The computation time for six double integrator agents, W R 8 8 and a second order dynamic feedback law is s, where randomly generated starting points to find a stabilizing initial controller have been considered. This can be reduced if an initial stabilizing controller is provided to the optimization routine. Note that any stabilizing controller guarantees convergence of the double integrator All simulation examples as well as the software to design fixedorder controllers with replicated structure are available at the web site: TABLE I CONVERGENCE TIMES FOR DIFFERENT CONTROLLER SYNTHESIS METHODS FOR THE DOUBLE INTEGRATOR MODEL Controller Local controller order T c[s] u max Optimal Weights [].68 Static Controller 9.7 Dynamic Controller # 6.9 Dynamic Controller # 7. model since all eigenvalues of Ξ except for the eigenvalue at one which cannot be influenced by the feedback law lie in the closed unit disc D which guarantees consensus for the double integrator model according to Lemma. A dynamic controller referred to as controller # with n K = is designed to show a behavior similar to the optimal static feedback law, whereas the dynamic controller # which has the same number of states as controller # is designed for very fast convergence at the expense of an increased control effort. Fig. shows the square root of the sum (over all agents) of squares of the deviation from the consensus value for the positions of the agents and Table I gives an overview on the convergence time of the agents where it is assumed that the convergence time T c [s] is the longest time for all agents to reach an interval of. units from the desired average consensus velocity. Table I shows that the optimal convergence time reported in [] is almost attained by the proposed approach when using a static controller (). For higher order control laws like (), one can shape the controller to improve convergence time or find a compromise between convergence time and control effort. N (ζi ζ) i= Optimal Weights [] Static Controller Dynamic Controller # Dynamic Controller # Fig.. Deviation from average consensus value ζ for the double integrator model B. High-Order Multi-Agent Systems To iluustrate the flexibility of the proposed controller design method, the approach is also applied to formation flight of quad-rotor helicopters where a th order linear statespace model given in [6], [] is used to model the quad- 67

5 Positions 8 6 (a) Positions 8 6 (b) Positions 8 6 (c) 8 Positions 6 (d) Fig.. Comparison of static and dynamic feedback laws for the double integrator model, (a) network with optimal parameters and feedback law (), see [], (b) network designed with proposed method and feedback law (), (c)-(d) network designed with proposed method and nd order dynamic feedback law rotors. Time-domain results of a formation flight scenario are presented in Fig. 6. In this simulation scenario, a MAS with N = quad-rotor helicopters is considered, where a constant undirected ring communication topology is assumed. The group starts at x() = [x () x () x () x () x ()] T = [m m m m m] T and is supposed to attain a formation where every agent has a difference of m in x- direction to the next agent. Note that there is no leader agent, which prevents the formation from reaching an absolute reference position. An output disturbance d y = σ(t t d ), t d = s, acts on agent no.. In Fig. 6 the x-coordinates of all agents are shown for a robust design approach presented in [6] compared with the simultaneous design of communication weights and local controller presented in this paper. In [6], the controller is designed such that it stabilizes every communication topology when all weights are set to one, and therefore inherits more conservatism. In both simulations the controller is restricted to be static, see Section III-B. V. CONCLUSIONS This work proposes a simultaneous design of communication weights and local controller to enhance the flexibility in the design procedure and to improve performance of a MAS. The controller synthesis is performed using a gradientbased method which is capable of dealing with fixed-order and replicated controller structures. The proposed method is applied to minimize the convergence time while at the same time restricting the control effort of a MAS consisting of double integrator agents. A formation control problem of 67

6 Simulation of a formation flight Simulation of a formation flight x positions x positions Fig. 6. Comparison between robust design technique (left) and simultaneous design of communication weights and local controller (right) for the quad-rotor helicopter model quad-rotor helicopters is also considered, where it is shown that for a moderate number of agents, including the freedom of optimizing the weights of the communication topology increases the performance. Current research is aiming at extending this approach to directed communication topologies. REFERENCES [] R. Olfati-Saber, J. A. Fax, and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, vol. 9, no., pp., 7. [] W. Ren, R. W. Beard, and E. M. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Systems Magazine, vol. 7, no., pp. 7 8, 7. [] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, vol. 9, no. 9, pp. 6 76,. [] G. Lafferriere, A. Williams, J. S. Caughman, and J. J. P. Veerman, Decentralized control of vehicle formations, Systems & Control Letters, vol., no. 9, pp ,. [] P. Massioni and M. Verhaegen, Distributed control for identical dynamically coupled systems: A decomposition approach, IEEE Transactions on Automatic Control, vol., no., pp., 9. [6] J. Wang and N. Elia, Agents design for distributed consensus over networks of fixed and switching topologies, in Proc. Joint 8th IEEE Conference on Decision and Control and 8th Chinese Control Conference, Shanghai, P.R. China, 9, pp [7] Y. Liu and Y. Jia, H consensus control of multi-agent systems with switching topology - a dynamic output feedback protocol, International Journal of Control, vol. 8, no., pp. 7 7,. [8] L. Xiao and S. Boyd, Fast linear iterations for distributed averaging, System and Control Letters, vol., pp. 6 78,. [9] A. Olshevsky and J. Tsitsiklis, Convergence speed in distributed consensus and averaging, SIAM Review, vol., no., pp ,. [] W. Ren, On consensus algorithms for double integrator dynamics, IEEE Transactions on Automatic Control, vol., no. 6, pp. 9, 8. [] D. W. Casbeer, R. Beard, and A. L. Swindlehurst, Discrete double integrator consensus, in 7th IEEE Conference on Decision and Control, 8, pp [] J. Zhu, On consensus speed of multi-agent systems with doubleintegrator dynamics, Linear Algebra and its Applications, vol., no., pp. 9 6,. [] A. Eichler and H. Werner, Closed-form solution for optimal convergence speed of multi-agent systems with discrete-time doubleintegrator dynamics, Systems & Control Letters,, submitted. [Online]. Available: rtsae/rtsae SCL.htm [] J. Burke, D. Henrion, A. Lewis, and M. Overton, HIFOO - a Matlab package for fixed-order controller design and H optimization, in IFAC Symposium on Robust Control Design, Toulouse, France, 6. [] R. Diestel, Graph Theory, th ed., ser. Graduate Texts in Mathematics. Springer Verlag,, vol. 7. [6] U. Pilz, A. Popov, and H. Werner, Robust controller design for formation flight of quad-rotor helicopters, in Proc. Joint 8th IEEE Conference on Decision and Control and 8th Chinese Control Conference, Shanghai, P.R. China, 9, pp [7] G. P. Liu, Nonlinear Identification and Control - A Neural Network Approach, st ed., ser. Advances in Industrial Control - Lecture Notes in Control and Information Sciences. Springer-Verlag New York, Inc.,. [8] C. A. Coello Coello, G. B. Lamont, and D. A. Veldhuizen, Evolutionary Algorithms for Solving Multi-Objective Problems, nd ed., ser. Genetic and Evolutionary Computation. Springer-Verlag New York, Inc., 7. [9] P. Apkarian and D. Noll, Nonsmooth H synthesis, IEEE Transactions on Automatic Control, vol., no., pp. 7 86, 6. [] S. Gumussoy, M. Millstone, and M. Overton, H strong stabilization via HIFOO, a package for fixed-order controller design, in Proc. 7th IEEE Conference on Decision and Control, Cancun, Mexico, 8, pp.. [] A. Popov, H. Werner, and M. Millstone, Fixed-structure discrete-time H controller synthesis with HIFOO, in Proc. 9th IEEE Conference on Decision and Control,, pp.. [] A. Popov and H. Werner, H controller design for a multi-agent system based on a replicated control structure, in Proc. of American Control Conference, Baltimore, MD, USA,, pp. 7. [] T. Iwasaki and R. E. Skelton, All controllers for the general H control problem: LMI existence conditions and state space formulas, Automatica, vol., no. 8, pp. 7 7, 99. [] D. Lara, A. Sanchez, R. Lozano, and P. Castillo, Real-time embedded control system for VTOL aircrafts: application to stabilize a quad-rotor helicopter, in IEEE Conference on Control Applications, Munich, Germany, 6, pp